Review(cont.) Use binomial, multinomial and Poisson distribution Not normality distribution Tow most used models: logistic regression(logit) log linear
Something unfamiliar Maximum likelihood estimation Confidence intervals Statistical inference for binomial parameters multinomial parameters ……
Terminology and notation Cell Contingency table
Terminology and notation Subjective Sensitivity and Specificity Conditional distribution Joint distribution Marginal distribution Independence =>
Sampling Scheme Poisson the joint probability mass function: Multinomial independent/product multinomial Hyper geometric
Example for sampling
Types of studies Retrospective: case-control Prospective: – Clinical trial observational study – Cohort study Cross-sectional: experimental study
Comparing two proportions Difference Relative risk Odds ratio – Odds defined as – For a 2*2 table, odds ratio – Another name: cross-product ratio
Properties of the Odds Ratio 0=<θ <, θ=1 means independence of X and Y the farther from 1.0, the stronger the association between X and Y. log θ is convenient and symmetric Suitable for all direction No change when any row/column multiplied by a constant.
Partial association in stratified 2*2 tables Experimental studies We hold other covariates constant to study the effect of X on Y. Observational studies Control for a possibly confounding variable Z Partial tables=>conditional association Marginal table
Death penalty example
Death penalty example(cont.)
Conditional and marginal odds ratios Conditional Marginal
Marginal versus Conditional(cont.) Marginal conditional
Homogeneous Association For a 2*2*K table, homogeneous XY association defined as: A symmetric property: – Applies to any pair of variables viewed across the categories of the third. – No interaction between two variables in their effects on the other variable.
Homogeneous Association(cont.) Suppose: – X=smoking(yes, no) – Y=lung cancer(yes, no) – Z=age( 65) – And Age is an Effect Modifier
Extensions for i*j Tables For a 2*2 table Odds ratio An i*j table Odds ratios
Representation methods Method 1
For I*J tables (I-1)*(J-1) odds ratios describe any association All 1.0s means INDEPENDENCE! Three-way I*J*K tables, Homogeneous XY association means: any conditional odds ratio formed using two categories of X and Y each is the same at each category of Z.
Measures of Association Two kinds of variables: – Nominal variables – Ordinal variables Nominal variables: Set a measure for X and Y: – V(Y),V(Y|X) Proportional reduction:
Measures of variation Entropy: Goodman and Kruskal(1954) (tau) Lambda:
About Entropy Uncertainty coefficient: U=0=>INDEPENDENCE U=1=>π(j|i)=1 for each i, some j. Drawbacks: No intuition for such a proportional reduction.
Ordinal Trends An example:
Three kinds of relationship Concordant Discordant Tied
Example(cont.) D=849 Define (C-D)/(C+D) as Gamma measure. Here, A weak tendency for job satisfaction to increase as income increases.
Properties of Gamma Measure Symmetric Range [-1,1] Absolute value of 1 means perfect linear Monotonicity is required for Independence =>,not vice-versa.